All polynomials with positive integer coefficients of degree 0 are in S since a_0 = 1+ (a_0-1)*1, i.e letting c=a_0-1, f=g=1 . For all k, x^k is in S since x is in S and if x^i is in S then x^{i+1}=x.x^i .

If all polynomials of degree n are in S, then any polynomial of degree n+1 can be written as f + a_{n+1} x^{n+1}, where f is of degree n, and letting c=a_{n+1} and g=x^{n+1} this is seen to be in S too.